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1. • CommentRowNumber1.
   • CommentAuthorGuillaume Brunerie
   • CommentTimeNov 25th 2011
   • PermaLink
   Author: Guillaume Brunerie
   Format: MarkdownItexHi all, I'm reading [spectrum](http://ncatlab.org/nlab/show/spectrum) and [suspension spectrum](http://ncatlab.org/nlab/show/suspension+spectrum) and I'm very confused. In the first page, an $\Omega$-spectrum is defined as a sequence of pointed spaces $E_n$ indexed by the natural numbers, together with isomorphisms $E_n\simeq{}\Omega{}E_{n+1}$. In the second page, given a based space $X$, its suspension spectrum is defined as the $\Omega$-spectrum $E$ with $E_n=\Sigma^n X$. But there is absolutely no reason that $\Omega\Sigma^{n+1}X\simeq\Sigma^n X$, so this is not an $\Omega$-spectrum. Unfortunately, I'm not sure what (and how) this should be changed (I came here to learn what is a spectrum). I've seen at several other places a spectrum being defined as a family of spaces together with maps $\Sigma{}E_n\to\E_{n+1}$, with $\Omega$-spectra being those spectra where the associated map $E_n\to\Omega{}E_{n+1}$ is an homotopy equivalence (and in this case, it seems that the suspension spectrum of a topological space does exists, but is not an $\Omega$-spectrum). Could someone that know more about this stuff clarify the definitions and examples? Thanks! PS : Why does the LaTeX looks much better in the nLab than here?

   Hi all,

   I’m reading spectrum and suspension spectrum and I’m very confused.

   In the first page, an $\Omega$-spectrum is defined as a sequence of pointed spaces $E_n$ indexed by the natural numbers, together with isomorphisms $E_n\simeq{}\Omega{}E_{n+1}$. In the second page, given a based space $X$, its suspension spectrum is defined as the $\Omega$-spectrum $E$ with $E_n=\Sigma^n X$. But there is absolutely no reason that $\Omega\Sigma^{n+1}X\simeq\Sigma^n X$, so this is not an $\Omega$-spectrum.

   Unfortunately, I’m not sure what (and how) this should be changed (I came here to learn what is a spectrum). I’ve seen at several other places a spectrum being defined as a family of spaces together with maps $\Sigma{}E_n\to\E_{n+1}$, with $\Omega$-spectra being those spectra where the associated map $E_n\to\Omega{}E_{n+1}$ is an homotopy equivalence (and in this case, it seems that the suspension spectrum of a topological space does exists, but is not an $\Omega$-spectrum).

   Could someone that know more about this stuff clarify the definitions and examples?

   Thanks!

   PS : Why does the LaTeX looks much better in the nLab than here?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2011
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   Author: Urs
   Format: MarkdownItexYou are right, there is a mix-up in terminology. I'll try to fix it a little later when I have a minute.

   You are right, there is a mix-up in terminology. I’ll try to fix it a little later when I have a minute.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeNov 26th 2011
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   Author: Mike Shulman
   Format: MarkdownItexThe quick answer is that there is a model structure on the spectra defined in terms of maps $\Sigma E_n \to E_{n+1}$ (Peter May calls those "prespectra"), for which the $\Omega$-spectra are the fibrant objects. So the $\Omega$-spectra are in some sense the "real" spectra, just as Kan complexes are the "real" $\infty$-groupoids --- if you want to get the "real" suspension spectrum of a space, you have to take a fibrant replacement of the direct construction.

   The quick answer is that there is a model structure on the spectra defined in terms of maps $\Sigma E_n \to E_{n+1}$ (Peter May calls those “prespectra”), for which the $\Omega$-spectra are the fibrant objects. So the $\Omega$-spectra are in some sense the “real” spectra, just as Kan complexes are the “real” $\infty$-groupoids — if you want to get the “real” suspension spectrum of a space, you have to take a fibrant replacement of the direct construction.

4. • CommentRowNumber4.
   • CommentAuthorjim_stasheff
   • CommentTimeNov 26th 2011
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   Author: jim_stasheff
   Format: TextI think the question referred to a linguistic problem, which might be avoided by strict definitions of spectra, Ω-spectra and suspension spectra.

   I think the question referred to a linguistic problem, which might be avoided by strict definitions of spectra, Ω-spectra
   and suspension spectra.
5. • CommentRowNumber5.
   • CommentAuthorGuillaume Brunerie
   • CommentTimeDec 9th 2011
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   Author: Guillaume Brunerie
   Format: MarkdownItexI did a few modifications at the two pages to have something that look less wrong and I added what Mike said (thanks!) @Jim : it was both :-)

   I did a few modifications at the two pages to have something that look less wrong and I added what Mike said (thanks!)

   @Jim : it was both :-)